We report on a novel class of higher-order Bessel-Gauss beams in which the well-known Bessel-Gauss beam is the fundamental mode and the azimuthally symmetric Laguerre-Gaussian beams are special cases. We find these higher-order Bessel-Gauss beams by superimposing decentered Hermite-Gaussian beams. We show analytically and experimentally that these higher-order Bessel-Gauss beams resemble higher-order eigenmodes of optical resonators consisting of aspheric mirrors. This work is relevant for the many applications of Bessel-Gauss beams in particular the more recently proposed high-intensity Bessel-Gauss enhancement cavities for strong-field physics applications.
© 2012 Optical Society of America
Bessel-Gauss beams offer properties that Gaussian beams do not possess. For instance, they exhibit an extended depth of field, and away from a strong on-axis intensity region they exhibit an annular intensity distribution in the far field [1, 2]. These attributes have made these beams interesting for applications in biomedical imaging, material processing, particle trapping, or laser-based acceleration of particles [3–7].
Laser resonators supporting Bessel-Gauss beams have also been designed and implemented [8–10]. More recently, enhancement cavities based on Bessel-Gauss-type modes have been proposed for strong-field applications, such as cavity-enhanced high-harmonic generation (HHG). Bessel-Gauss modes offer the benefits of near-perfect out-coupling of the generated intra-cavity high-harmonics and an increased intensity ratio from the focus to the cavity mirror surfaces . In these applications the well-known Bessel-Gauss beam has been exclusively considered as the operating mode. Having observed higher-order azimuthally symmetric mode solutions with numerical mode-solvers, the most pertinent questions we ask are: Do higher-order Bessel-Gauss beams exist? How can they be described analytically and demonstrated experimentally?
The existence of higher-order Bessel-Gauss beams showing azimuthal phase dependence (i.e. carry optical angular momentum) is known [12, 13]. However, azimuthally symmetric higher-order Bessel-Gauss beams are hitherto not reported in the literature, to the best of our knowledge.
In the present paper, we first analytically describe higher-order Bessel-Gauss beams. Based on these findings, we then design an optical resonator and show numerically that it supports modes resembling azimuthally symmetric higher-order Bessel-Gauss beams. Finally, we demonstrate these higher-order Bessel-Gauss modes experimentally.
2. Generalizing higher-order Bessel-Gauss beams
Conventional Bessel-Gauss beams can be built up by a superposition of decentered Gaussian beams, whose centers are positioned on a circle and whose beam directions are pointing to the apex of a cone [14, 15]. In the following we will show that higher-order Bessel-Gauss beams can be formed in a similar way by using the more general decentered Hermite-Gaussian beams as component beams for the superposition on a single cone.
First, we define decentered Hermite-Gaussian beams and describe their propagation through optical systems. After that we superpose these component beams to the higher-order Bessel-Gauss beams and also discuss their propagation properties. Finally, we show that these beams can collapse to the azimuthally symmetric Laguerre-Gaussian beams.
2.1. Decentered Hermite-Gaussian beams
Decentered Hermite-Gaussian beams are a generalization of the conventional Hermite-Gaussian beam solution. The additional features are a displacement of their beam center by a vector rd0 = (xd0, yd0) in the x–y plane and a tilt of the mean beam direction with respect to the z direction by an angle ε0. They are also a solution to the paraxial wave-equation, and we explicitly define the decentered Hermite-Gaussian beam at the input of an optical system (z=0) as followsFig. 1.
2.2. Propagation of decentered Hermite-Gaussian beams through optical systems
The paraxial propagation of the decentered Hermite-Gaussian beam through an ABCD optical system is described by the Collins integral . By exploiting the analogy to the solution for the conventional Hermite-Gaussian beam , we obtain for the final state of the propagation of the decentered Hermite-Gaussian beam:
The decentered Hermite-Gaussian beams are formulated as a function of propagation parameters, i.e. the components of the ABCD matrix, in order to derive a propagation-dependent solution for the higher-order Bessel-Gauss beams in the next section.
2.3. Superposing decentered Hermite-Gaussian beams to higher-order Bessel-Gauss beams
We form the higher-order Bessel-Gauss beams by superimposing the component beams with the following attributes: the Hermite-Gaussian component is modulated only in the radial direction. The vectors rd0 and ε0 are collinear with rd0= const. and ε0= const. This corresponds to a superposition of beams whose mean centers are placed on a circle with radius rd0 in the z=0 plane and whose mean beam directions point to the apex of a single cone with a semi-aperture angle ε0, as shown in Fig. 1.
For the superposition, we express each component beam in an auxilliary coordinate system, which is rotated by an angle γ with regard to a fixed reference coordinate system (r,θ,z) so that its abscissa points in the direction of the vector rd0. In this auxiliary system, the decentered Hermite-Gaussian beam consists solely of the Hm-term (i.e. Hn=0 = 1) and the abscissa coordinate is given by r·cos(θ−γ). The propagated decentered Hermite-Gaussian beam is expressed in the cylindrical coordinates (r,θ,z) asEqs. (5) and (6). The formulation of higher-order Bessel-Gauss beams in terms of ABCD-matrix parameters allows us to study their transformation under the impact of optical components. We will expand upon this feature in section 3.1 describing optical resonators for these beam solutions.
The generalization of the Bessel-Gauss beam solution is found by integrating over the γ angle from 0 to 2π:Appendix), and the first four expressions are given by 14]. Furthermore, the solutions comprising the terms ℐ0l (i.e. m = 0) are identical to previously defined higher-order Bessel-Gauss beams exhibiting an azimuthal phase-variation [12, 13].
Here, we introduce higher-order Bessel-Gauss beams with a radial index m > 0. In Fig. 2 we illustrate the effect of the radial index by plotting the azimuthally symmetric (l = 0) higher-order Bessel-Gauss beams for different orders m in the near and far-field. The parameters are w0 = 200μm, rd0 = 0, ε = 0.3°, and λ = 1040nm. In the far-field (here, calculated at z= 4 · zR, where ), all beams show an annular beam with variations in the radial direction such that the number of nodes corresponds to m. If l > 0 these solutions exhibit azimuthal phase-variations, as displayed in Fig. 3. To highlight the impact of the azimuthal index l on the solution, we consider the superposition (um,l + um,−l) resulting in a cosine-variation over the angular coordinate.
How does the Bessel-Gauss beam solution depend on the parameters rd0 and ε? - A classification is known for the fundamental Bessel-Gauss beam : Generalized Bessel-Gauss beam, ordinary Bessel-Gauss beam and modified Bessel-Gauss beam denote the cases (rd0 ≠ 0, ε0 ≠ 0); (rd0 = 0, ε0 ≠ 0), and (rd0 ≠ 0, ε0 = 0), respectively. We extend this terminology to the higher-order Bessel-Gauss beams. For the example of a Bessel-Gauss beam with m=2, we plot the different cases in Fig. 4. On the left side, the radial profile is plotted over the propagation distance. On the right side, we display the transverse beam pattern at the position of minimum Gaussian waist (here at z=0). Figures 4(a) and 4(b) show the absolute value of the amplitude for a generalized higher-order (m=2) Bessel-Gauss beam for w0 = 200μm, rd0 = 0.8mm, ε = 0.3°, and λ = 1040nm. The vertical blue line highlights the (z=0) position. For this beam type the intersection with the symmetry axis is shifted from the position where the minimum Gaussian waist occurs. Figures 4(c) and 4(d) displays the ordinary beam type. The intersection coincidences with the position of minimum waist. Figures 4(e) and 4(f) picture the modified beam type. Furthermore, if both rd0 and ε0 are zero, a azimuthally symmetric Laguerre-Gaussian beam is recovered, as displayed in Fig. 4(g) and 4(h).
2.4. Collapse of the solution to Laguerre-Gaussian beams for ε = 0 and rd0 = 0
To show the collapse of the azimuthally symmetric higher-order Bessel-Gauss beams to Laguerre-Gaussian beams, we write the integral in Eq. (12) for ε = 0 and rd0 = 0. For uneven m the integral vanishes due to symmetry, and for even m (= 2 · p) we find:18]:
Equation (20) describes the remarkable property that the azimuthally symmetric Laguerre-Gaussian beams are a subset of the presented novel beam class. Thus, the presented solutions are a more general description for azimuthally symmetric beams.
3. Comparing the Bessel-Gauss beam solutions to modes of optical resonators
In the following, we would like to compare the analytical expressions for the fundamental and azimuthally symmetric higher-order Bessel-Gauss beams with numerical results. Specifically, we simulate a resonant optical cavity that supports a conventional generalized Bessel-Gauss beam for the fundamental mode, and should exhibit higher-order Bessel-Gauss beams as higher-order modes.
3.1. Design of the optical resonator
Figure 5 shows a schematic of the optical resonator, it consists of a spherical mirror with a radius of curvature R and an axicon mirror with a base angle ε.
Ray matrices can be employed for the design of the optical resonator. One roundtrip is described byEqs. (8) and (9). At the axicon mirror, these parameters are described by (rd0, ε) and (rd0, −ε) before and after one roundtrip, respectively. Thus, the resonator length is given by
To summarize, the resonator length and the waist w0 of the Gaussian component of the Bessel-Gauss beam solution are fully governed by the radius of curvature of the spherical resonator mirror, the base angle of the axicon mirror, as well as the radius of the ring of the generalized Bessel-Gauss beam on the axicon, rd0. For the following calculations, we chose a radius of curvature R = 250 mm, ε = 0.5° and rd0= 1.5 mm, which result in L ≈ 78 mm and w0 ≈ 196μm for a light wavelength of 1040 nm. These particular parameters were selected since they approximate the experimental configuration which, in turn, was determined by the availability of base angles for the axicon mirror and radii of curvatures for the spherical mirror. It results in a ring radius of the Bessel-Gauss beam that avoids the round tip on the axicon mirror and results in a not too large ring radius on the curved mirror. This situation is advantageous as it reduces effects of surface variations on the resonator mode.
The spherical mirror serves as the coupler to the resonator. For the coupling to the resonator with an excitation beam, it is important to know the beam parameters at the plane backside of this mirror. Geometrical optics of the meridional ray, characterizing the propagation of the Bessel-Gauss mode, gives a semi-aperture angle ε′ ≈ nm · ε, where nm is the refractive index of the mirror substrate, and a ring radius r′d ≈ rd + Δ · ε, where the ring-radius at the front surface is given by rd = ε · R and Δ is the geometrical thickness of the mirror, which is given by Δ = Δout − s(rout), where Δout is the mirror thickness at the outer radius rout and the surface function is given by .
3.2. Numerical mode solver
Diffraction can be described in terms of a Hankel transform, which we numerically implement by employing a quasi-discrete Hankel transform (i.e. a Fourier-Bessel series expansion) [19–21]. In the following, we solve for the azimuthally symmetric modes.
To robustly obtain the numerical results for the higher-order modes, we have developed a fast two-stage mode solver. In the first stage, the modes are guessed by the principal component analysis (PCA) and in the second stage, a Fox-Li method refines these inputs to the actual mode solution. In particular, a generalized (fundamental) Bessel-Gauss beam is used as the start for the first stage. However, we intentionally use a larger waist w0 and ring radius rd0 than the predicted analytical solution for the fundamental mode would possess. This beam is propagated for 20 iterations. A matrix is formed by the roundtrip vectors. The PCA looks for the least correlated vectors, so that a linear combination of these so called principal components recovers the original data. Thus, the principal components are similar to the mode shapes, and serve as the inputs for the Fox-Li algorithm, which typically requires several hundreds of iterations. Here, we use only 500 iterations. The mode is then determined by looking for the coherent resonance of the sum of the iterated fields, for this the n-th iterated field is multiplied by the n-th power of a phasor in which the phase-shift is varied between 0 and 2·π . To calculate the next higher-order mode, the second stage is restarted with a different higher-order principal component.
3.3. Comparison of the numerical mode-solution to the Bessel-Gauss beam solutions
To demonstrate the accuracy of the analytical expressions, the Bessel-Gauss beam solutions of different radial orders m are evaluated and compared to numerical results at the axicon mirror, and at two z-positions outside the cavity. The mode field is propagated in free space from the axicon mirror to the on-axis crossing point at a distance −(R − L) via an intermediate position −(0.9 · R − L). In Fig. 5, these positions are denoted as P1, P2 and P3. Figure 6 shows a comparison between numerical results and the analytical solutions. It can be seen that the azimuthally symmetric higher-order Bessel-Gauss modes represent an accurate analytical model for higher-order modes in resonators supporting the (m=0, l=0) Bessel-Gauss beam as the fundamental mode. Conversely, the numerical results validate the existence of azimuthally symmetric higher-order Bessel-Gauss beams, and constitute a reference for the hitherto unknown analytical beam solution.
Figure 7 shows the free-space propagation of Bessel-Gauss beams of different radial orders m. The propagation starts at the axicon mirror (z=0), which corresponds to the minimum waist point of the Gaussian component of the generalized Bessel-Gauss beams, and evolves towards the on-axis crossing point (without considering refraction in the axicon). It can be seen that for higher-order Bessel-Gauss beams the amplitude variation on the annulus maps into an on-axis intensity variation. Additionally, the length of this on-axis, modulated depth-of-field increases with radial order m. Specifically, its range can be approximated as Δz ∼ wm/ sin (ε0) , where wm stands for the effective mode size of the m-th order Hermite-Gaussian, which is related to the Gaussian waist as . For a generalized Bessel-Gaussian beam, we find an approximate expression by evaluating the Gaussian waist at the geometrical on-axis crossing point zc = rd0/ tan (ε0), i.e. w = w0 · (1+(zc/zR)2)(1/2), and for an ordinary Bessel-Gauss beam w = w0 since zc = 0. To adjust the on-axis intensity for laser applications, several parameters of the azimuthally symmetric higher-order Bessel-Gauss beams can be adjusted. Key beam parameters are the mode order m, the waist w0, the semi-aperture angle ε and the ring radius at minimum waist rd0.
3.4. Experimental setup
To observe the higher-order Bessel-Gauss beams experimentally, we have designed an enhancement cavity according to the guidelines, which have been presented in section 3.1. As shown in Fig. 5, the resonator is formed by an axicon mirror (AM) and a spherical mirror (SM). In the experiment, the conic side of the axicon mirror has a base angle of 0.5° and the coating has a reflectance of 0.99. The spherical mirror has a radius of curvature of R = 250 mm and a reflectance of 0.9. The 1-inch diameter spherical mirror (with a 1/4-inch thickness of the BK-7 substrate) serves as the input coupler to the resonator. The back sides of both mirrors are anti-reflection coated. In the design configuration, the separation distance between the two mirrors is L = 75 mm, and the ring radius of the Bessel-Gauss mode on the axicon is approximately 1.5 mm.
A schematic of the excitation path to the resonator is shown in Fig. 8. We use a tunable grating-stabilized external cavity diode laser, which is coupled to a single-mode fiber (PM980-XP, mode-field diameter of approximately 6.6 μm). The laser is operated at a wavelength of 1040 nm and it emits up to 30 mW of power at the fiber output with a line width of 100 kHz. The laser frequency is varied by tuning the grating angle of the external cavity with a piezoelectric element. This allows us to lock the laser frequency to the resonance of the enhancement cavity by dither locking. For this, the light that is reflected from the in-coupling mirror SM is detected at the side port of an optical isolator with a photodiode and is used as a feedback signal for the locking loop. The leakage through the axicon mirror is used to record the resonator modes with a camera, which is placed behind AM.
To match the laser beam to the resonator mode, we first magnify the beam emanating from the fiber by a factor of 450 using the lenses L1, L2, and L3 with focal lengths of 4 mm, 10 mm, 100 mm, respectively. An axicon (A) generates an approximate Bessel-Gauss beam with a semi-aperture angle of (n − 1) · δ = 0.45°, where n is the refractive index of fused silica (n=1.45) and δ the base angle of the axicon, δ = 1°. The waist of the beam illuminating the axicon is 1.5 mm. Due to the round, and thus imperfect, apex of the axicon, the resulting beam in the far field is not a pure ring-like beam but still contains unwanted light in the central region. The quality of the beam is increased by employing Fourier filtering with an opaque disk of 1 mm radius in a 4-f system (150 mm focal-length lenses L4 and L5) . Behind the 4-f system, a 300 mm focal-length lens (L6) and a 200 mm focal-length lens (L7) match the excitation beam to the beam parameters required at the plane side of SM.
To determine the positions of the lenses L6 and L7, at first, we consider a configuration that assumes an ideal Bessel-Gauss beam being produced by the axicon. After the 4-f system the excitation beam is collimated by the lens L6 and is then focused into the resonator by the lens L7. This results in a semi-aperture angle of 3/2· 0.45° = 0.68°. The ring radius of the beam on SM determines the distance between L7 and SM. To better match the beam to the cavity mode, the excitation path is simulated numerically. We account for an imperfect axicon showing a parabolic shape that deviates by 8 μm from the ideal conic shape at the tip. As a consequence, we iteratively improve the distances between L5 and L6, L6 and L7, as well as L7 and SM for best excitation. The resulting values are given in parentheses in Fig. 8, and are used as a starting point for the experiment, in which we also adjust the resonator mirrors and set the final positions of L6 and L7 so that the photodiode signal shows the most pronounced resonance feature.
3.5. Experimental results: comparison of the experimental modes to the beam solutions
Figure 9 shows the photodiode signal during a scan of the laser frequency. The resonances of the higher-order modes are much less pronounced compared to the one of the fundamental Bessel-Gauss mode. This is due to the lower excitation efficiency of the higher-order beam with the beam coupled into the cavity. However, the resonance can be still recognized in the photodiode signal and images of the resonator modes can be taken, as shown in Fig. 10(a)–(d). To obtain these pictures, we limit the range of the frequency scan to the resonance region of the mode of interest. The images are recorded at a scan rate of the piezo that is fast compared to the integration time of the camera. In this way, the camera averages several scans. These images were taken at a distance z= 25 mm behind the flat surface of the axicon mirror. The evolution of the beam profile as a function of other distances from the axicon mirror will be similar to the one shown in Fig. 7. However, it must be noted that in Fig. 7 we have neglected refraction effects by the axicon-mirror substrate. Behind the axicon the semi-aperture angle of the higher-order Bessel-Gauss mode is changed from 0.5° to 0.725° due to refraction. Compared to Fig. 7, this will move the on-axis crossing-point of the beam closer to the axicon mirror. In Fig. 10 azimuthal variations of the intensity on the annulus can be seen, which are due to surface imperfections of the resonator mirrors . Faint central rings can be observed in the experimental data for the higher-order modes, Fig. 10 (b), 10(c) and 10(d). These artifacts are due to transmitted excitation light. The main reason for the show up are the weak resonances of the higher order modes compared to the fundamental mode, as shown in Fig. 9. Figures 10(e)–(h) shows numerical simulations corresponding to the experimental data (taking the refraction of the beam in the axicon mirror substrate into account).
In this work, Bessel-Gauss beams have been generalized to include a set of higher-order solutions that are analytically derived and experimentally demonstrated. Expressions for these beams have been obtained by superposing decentered Hermite-Gaussian beams. The well-known Bessel-Gauss beam is the fundamental solution of this class. Moreover, we have rigorously demonstrated that the azimuthally symmetric Laguerre-Gaussian beams are special cases of these higher-order Bessel-Gauss beams. Thus, these formerly distinct beam types i.e. Bessel-Gauss beams and Laguerre-Gauss beams have been unified. Furthermore, we have shown that the generalized higher-order Bessel-Gauss beams are present as higher-order modes in optical resonators consisting of aspheric mirrors.
Enhancement cavities based on Bessel-Gauss beams are seen to be an increasingly important way to employ strong-field physics in practical applications. Higher-order Bessel-Gauss beams can be used to tailor the intensity distribution in such devices. Other fields of application for these beams are imaging or control and manipulation of particles.
To solve the integral of Eq. (12), we write the Hermite-polynomials Hm in their explicit form. For this we use their recursion relation: H0(t) = 1, H1(t) = 2t and Hm+1(t) = 2tHm(t) − 2mHm−1(t). This results in an expansions in powers of cos-functions. The n-th power of the cos-functions can be written in terms of the harmonics of its argument by using De Moivre’s formula, e.g.:12]:
This work is supported by AFOSR grant FA9550-10-1-0063 and the Center for Free-Electron Laser Science. William P. Putnam acknowledges support by a NSF Graduate Fellowship.
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